Continuous maximal regularity on singular manifolds and its applications

Yuanzhen  Shao

doi:10.3934/eect.2016006

Article Contents

2016, Volume 5, Issue 2: 303-335. Doi: 10.3934/eect.2016006

This issue Previous Article New methods for local solvability of quasilinear symmetric hyperbolic systems Next Article A remark on blow up criterion of three-dimensional nematic liquid crystal flows

Continuous maximal regularity on singular manifolds and its applications

Yuanzhen Shao^1,

1.
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907

Received: December 2015

Revised: March 2016

Published: June 2016

Abstract / Introduction Related Papers Cited by

Abstract

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomenon for singular or degenerate parabolic equations can also be found in this paper.

Keywords:

Mathematics Subject Classification: 53C44, 58J99, 35K55, 35K65, 35K67, 35R01.

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